Integrand size = 14, antiderivative size = 104 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx=\frac {1}{2} \sqrt {3} b \sqrt [3]{c} \arctan \left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}-\frac {1}{2} b \sqrt [3]{c} \log \left (1-c^{2/3} x^2\right )+\frac {1}{4} b \sqrt [3]{c} \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right ) \] Output:
1/2*3^(1/2)*b*c^(1/3)*arctan(1/3*(1+2*c^(2/3)*x^2)*3^(1/2))-(a+b*arctanh(c *x^3))/x-1/2*b*c^(1/3)*ln(1-c^(2/3)*x^2)+1/4*b*c^(1/3)*ln(1+c^(2/3)*x^2+c^ (4/3)*x^4)
Time = 0.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.76 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx=-\frac {a}{x}+\frac {1}{2} \sqrt {3} b \sqrt [3]{c} \arctan \left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {1}{2} \sqrt {3} b \sqrt [3]{c} \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {b \text {arctanh}\left (c x^3\right )}{x}-\frac {1}{2} b \sqrt [3]{c} \log \left (1-\sqrt [3]{c} x\right )-\frac {1}{2} b \sqrt [3]{c} \log \left (1+\sqrt [3]{c} x\right )+\frac {1}{4} b \sqrt [3]{c} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{4} b \sqrt [3]{c} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right ) \] Input:
Integrate[(a + b*ArcTanh[c*x^3])/x^2,x]
Output:
-(a/x) + (Sqrt[3]*b*c^(1/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/2 - (Sqrt[ 3]*b*c^(1/3)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/2 - (b*ArcTanh[c*x^3])/x - (b*c^(1/3)*Log[1 - c^(1/3)*x])/2 - (b*c^(1/3)*Log[1 + c^(1/3)*x])/2 + (b* c^(1/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/4 + (b*c^(1/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/4
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6452, 807, 750, 16, 1142, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 3 b c \int \frac {x}{1-c^2 x^6}dx-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {3}{2} b c \int \frac {1}{1-c^2 x^6}dx^2-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \int \frac {1}{1-c^{2/3} x^2}dx^2+\frac {1}{3} \int \frac {c^{2/3} x^2+2}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \int \frac {c^{2/3} x^2+2}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2+\frac {\int \frac {c^{2/3} \left (2 c^{2/3} x^2+1\right )}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2}{2 c^{2/3}}\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2+\frac {1}{2} \int \frac {2 c^{2/3} x^2+1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {2 c^{2/3} x^2+1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2-\frac {3 \int \frac {1}{-x^4-3}d\left (2 c^{2/3} x^2+1\right )}{c^{2/3}}\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {2 c^{2/3} x^2+1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2+\frac {\sqrt {3} \arctan \left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{c^{2/3}}\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \left (\frac {\sqrt {3} \arctan \left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{c^{2/3}}+\frac {\log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{2 c^{2/3}}\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x}\) |
Input:
Int[(a + b*ArcTanh[c*x^3])/x^2,x]
Output:
-((a + b*ArcTanh[c*x^3])/x) + (3*b*c*(-1/3*Log[1 - c^(2/3)*x^2]/c^(2/3) + ((Sqrt[3]*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/c^(2/3) + Log[1 + c^(2/3)*x ^2 + c^(4/3)*x^4]/(2*c^(2/3)))/3))/2
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (c \,x^{3}\right )}{x}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{4}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}\) | \(105\) |
| parts | \(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (c \,x^{3}\right )}{x}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{4}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}\) | \(105\) |
| risch | \(-\frac {b \ln \left (c \,x^{3}+1\right )}{2 x}-\frac {a}{x}+\frac {b \ln \left (-c \,x^{3}+1\right )}{2 x}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(176\) |
Input:
int((a+b*arctanh(c*x^3))/x^2,x,method=_RETURNVERBOSE)
Output:
-a/x-b/x*arctanh(c*x^3)-1/2*b/c/(1/c^2)^(2/3)*ln(x^2-(1/c^2)^(1/3))+1/4*b/ c/(1/c^2)^(2/3)*ln(x^4+(1/c^2)^(1/3)*x^2+(1/c^2)^(2/3))+1/2*b/c/(1/c^2)^(2 /3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c^2)^(1/3)*x^2+1))
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx=-\frac {2 \, \sqrt {3} b \left (-c\right )^{\frac {1}{3}} x \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-c\right )^{\frac {2}{3}} x^{2} + \frac {1}{3} \, \sqrt {3}\right ) + b \left (-c\right )^{\frac {1}{3}} x \log \left (c^{2} x^{4} - \left (-c\right )^{\frac {1}{3}} c x^{2} + \left (-c\right )^{\frac {2}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {1}{3}} x \log \left (c x^{2} + \left (-c\right )^{\frac {1}{3}}\right ) + 2 \, b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a}{4 \, x} \] Input:
integrate((a+b*arctanh(c*x^3))/x^2,x, algorithm="fricas")
Output:
-1/4*(2*sqrt(3)*b*(-c)^(1/3)*x*arctan(2/3*sqrt(3)*(-c)^(2/3)*x^2 + 1/3*sqr t(3)) + b*(-c)^(1/3)*x*log(c^2*x^4 - (-c)^(1/3)*c*x^2 + (-c)^(2/3)) - 2*b* (-c)^(1/3)*x*log(c*x^2 + (-c)^(1/3)) + 2*b*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 4*a)/x
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(c*x**3))/x**2,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx=\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {2}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} + c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {2}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} - 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {2}{3}}}\right )} - \frac {4 \, \operatorname {artanh}\left (c x^{3}\right )}{x}\right )} b - \frac {a}{x} \] Input:
integrate((a+b*arctanh(c*x^3))/x^2,x, algorithm="maxima")
Output:
1/4*(c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(4/3)*x^2 + c^(2/3))/c^(2/3))/c^ (2/3) + log(c^(4/3)*x^4 + c^(2/3)*x^2 + 1)/c^(2/3) - 2*log((c^(2/3)*x^2 - 1)/c^(2/3))/c^(2/3)) - 4*arctanh(c*x^3)/x)*b - a/x
Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx=\frac {1}{4} \, b c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{{\left | c \right |}^{\frac {2}{3}}} + \frac {\log \left (x^{4} + \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{{\left | c \right |}^{\frac {2}{3}}} - \frac {2 \, \log \left ({\left | x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {2}{3}}}\right )} - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{2 \, x} - \frac {a}{x} \] Input:
integrate((a+b*arctanh(c*x^3))/x^2,x, algorithm="giac")
Output:
1/4*b*c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1/abs(c)^(2/3))*abs(c)^(2/3 ))/abs(c)^(2/3) + log(x^4 + x^2/abs(c)^(2/3) + 1/abs(c)^(4/3))/abs(c)^(2/3 ) - 2*log(abs(x^2 - 1/abs(c)^(2/3)))/abs(c)^(2/3)) - 1/2*b*log(-(c*x^3 + 1 )/(c*x^3 - 1))/x - a/x
Time = 5.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx=\frac {b\,\ln \left (1-c\,x^3\right )}{2\,x}-\frac {b\,c^{1/3}\,\ln \left (1-c^{2/3}\,x^2\right )}{2}-\frac {b\,\ln \left (c\,x^3+1\right )}{2\,x}-\frac {a}{x}-\frac {b\,c^{1/3}\,\ln \left (-\sqrt {3}-c^{2/3}\,x^2\,2{}\mathrm {i}-\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4}+\frac {b\,c^{1/3}\,\ln \left (-\sqrt {3}+c^{2/3}\,x^2\,2{}\mathrm {i}+1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4} \] Input:
int((a + b*atanh(c*x^3))/x^2,x)
Output:
(b*log(1 - c*x^3))/(2*x) - (b*c^(1/3)*log(1 - c^(2/3)*x^2))/2 - (b*log(c*x ^3 + 1))/(2*x) - a/x - (b*c^(1/3)*log(- 3^(1/2) - c^(2/3)*x^2*2i - 1i)*(3^ (1/2)*1i - 1))/4 + (b*c^(1/3)*log(c^(2/3)*x^2*2i - 3^(1/2) + 1i)*(3^(1/2)* 1i + 1))/4
Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^2} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 c^{\frac {1}{3}} x -1}{\sqrt {3}}\right ) b c x -2 \sqrt {3}\, \mathit {atan} \left (\frac {2 c^{\frac {1}{3}} x +1}{\sqrt {3}}\right ) b c x -4 c^{\frac {2}{3}} \mathit {atanh} \left (c \,x^{3}\right ) b -2 \mathit {atanh} \left (c \,x^{3}\right ) b c x -4 c^{\frac {2}{3}} a +2 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} x +1\right ) b c x -\mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) b c x -3 \,\mathrm {log}\left (c^{\frac {2}{3}} x -c^{\frac {1}{3}}\right ) b c x}{4 c^{\frac {2}{3}} x} \] Input:
int((a+b*atanh(c*x^3))/x^2,x)
Output:
(2*sqrt(3)*atan((2*c**(1/3)*x - 1)/sqrt(3))*b*c*x - 2*sqrt(3)*atan((2*c**( 1/3)*x + 1)/sqrt(3))*b*c*x - 4*c**(2/3)*atanh(c*x**3)*b - 2*atanh(c*x**3)* b*c*x - 4*c**(2/3)*a + 2*log(c**(2/3)*x**2 - c**(1/3)*x + 1)*b*c*x - log(c **(2/3)*x + c**(1/3))*b*c*x - 3*log(c**(2/3)*x - c**(1/3))*b*c*x)/(4*c**(2 /3)*x)